This invention relates generally to a special notation for symbolic logic and more particularly relates to useful and aesthetic devices for computing, teaching, demonstrating, and displaying the inner relationships in the algebra of logic, so that ordinary operations in symbolic logic can be performed with extreme calculational ease.
As is known to those skilled in the dicipline of symbolic logic, there are sixteen binary connective that exhaust all possible second-order true-false combinations for the two-valued logic of two sentences. As is also known to those skilled in this discipline, the performance of logical operations is limited by a number of inadequacies and complexities in the current notational systems. For instance, the most common notational system presently being used is the one historically evolved in the first decade of the 20th century by Peano-Whitehead-Russell, hereinafter called the PWR system.
In practice, the PWR system has several limitations. This system, for example, does not have a unique and easily recognizable symbol for each of the sixteen binary connectives (see TABLE I). The few symbols in use are a mixture of analogical and alphabetical shapes; they exist as separate entities--each as island unto itself. The interrelationships between these symbols are not immediately transparent, nor are they easily discernible by inspection; consequently, these interrelationships are not readily available to facilitate logical calculations. As a result, the symbols are cumbersome and abstract, and the interrelationships must be memorized before one is able to perform logical operations. Also, when these operations are performed, the PWR system lacks the ability to use a few, simple rules that apply uniformly when the symbols are made to act upon themselves. Furthermore, this prior system cannot be embodied or used in physical models that clearly display the interrelationships among these symbols. The absence of physical models is a decided disadvantage when it comes to teaching, demonstrating, and using the underlying structures of symbolic logic.
Another current notational system in use is the Polish system which employs a different, upper-case letter of the English alphabet to represent each of the sixteen binary connectives (see TABLE I). The alphabetical symbols of the Polish system, as in the PWR system, are also separate and isolated, which again puts a burden on memory and abstract thought, particularly when one wishes to consider the interrelationships among these symbols. Not designed to be in keeping with a few, simple, uniform rules of manipulation, the alphabetical symbols of the Polish system also cannot be acted upon directly to perform logical operations. The cumbersomeness with which interrelationships are handled in the Polish system also precludes this notation from being easily embodied in physical models for displaying, teaching, and performing logical operations.
A third notational system, one less prevalent than the previously discussed PWR and Polish systems, is the McCulloch system wherein each of the sixteen binary connectives is represented by an all-common X-frame (Greek letter "chi") to which is added the appropriate combination of dots in the four regions that surround the intersection at the center of the X-frame (see TABLE I). Although the logical meaning of each McCulloch symbol is geometrically and visually displayed, these symbols do not have distinct mnemonic values by which they can be easily identified. Also, this notation cannot be used unless the writer constantly repeats the X-frame, once for each symbol. In addition, the McCulloch system has not heretofore been used with a few, simple, uniform rules by which to flip and rotate the symbols themselves. Moreover, rules of this kind are available for these symbols only if the user is willing to endure a very awkward situation, perceptually.
As is also well-known by those skilled in the discipline of symbolic logic, there are over a dozen additional notational systems that are variations, for the most part modifications, extensions, or admixtures, of the three basic systems discussed above. In general, these variations in the notational systems include the several disadvantages listed above with respect to the three basic systems of PWR, Polish, and McCulloch.
In view of the above background and the existing notational systems used in symbolic logic, it is the object of this invention to devise a new notational system, namely, the logic alphabet, for which carefully combined features yield advantages that overcome the above noted disadvantages of the current notational systems.
Most of all, the logic alphabet of this invention is a direct continuation of, in some ways a completion of, the notational efforts of Charles Sanders Peirce (1839-1914). Expression of his efforts can be found in his Manuscripts 429, 431, and especially 530, as numbered in R. S. Robin's annotated catalogue of Peirce's papers. Perspective in regard to the works of Peirce is best obtained by way of the scholarly efforts of Max H. Fisch (1900- ), who is presently the General Editor of the Peirce Edition Project. This project entails the preparation of a new edition of Peirce's writings, expected to run to fifteen volumes. For an example of Professor Fisch's work, see the article entitled "Peirce's Arisbe: The Greek influence in his later life," which is found in the journal called "Transactions of the Charles S. Peirce Society" (1971, Volume 7, Pages 187-210).
First and foremost, the logic alphabet of this invention is a lesson in man-sign engineering. That is to say, the act of notation building is conducted in such a way that the new symbolism meets two requirements. First, it possesses mind-brain economies that fit the psychological characteristics of the person. Second, it possesses the same interrelatedness among its symbols that exists among the logical meanings being expressed. In effect, each symbol of the logic alphabet, like a small organism, is well adapted both to the society of people who will use it and to the society of symbols that will be used with it.
Fulfilling the requirements of man-sign engineering leads to a new approach. Unlike the decimal system used for numbers, which is a notation that is base consistent and value positional, the logic alphabet of this invention is a notation that is frame consistent and symmetry positional. That is to say, logical operations are performed by means of non-numerical motions, namely, by changing the positions of the symbols when symmetrically placing them in different orientations.
The logic alphabet of this invention, unlike the PWR system, is complete in the sense that a special set of shapes is assigned to all of the sixteen binary connectives, namely, certain shapes that are taken from the lower-case letters of the English alphabet (see TABLE I). Unlike the Polish system, the logic alphabet of this invention also gives a central role to some geometric properties that are, by careful selection, an inherent part of the letter-shapes themselves. In addition, unlike the McCulloch system, the logic alphabet of this invention is phonetic. Each symbol is identified with a distinct associational and mnemonic sound value, in most cases the same one that is assigned to the corresponding letter of the alphabet in normal use for reading. Consequently, the combined advantages of the logic alphabet of this invention not only go beyond the fragmentary advantages of the PWR system, but also give the geometric advantages of the McCulloch system to a Polish-like system, and conversely, give the phonetic advantages of the Polish system to a McCulloch-like system.
A key consideration is how best to think about the fact that the logic alphabet of this invention is constructed from symbols that are letter-shapes. Heavily weighted toward opposite extremes, the Polish and McCulloch systems are split-brain notations. That is to say, the Polish system favors the alphabetic and algebraic side, the letter side of letter-shapes; and the McCulloch system favors the iconic and geometric side, the shape side of letter-shapes. The PWR notation is so much in the fragmentary and analogical direction that it does not, in any systematic way, really participate in this distinction. In these terms, and by a contrast that grows out of a proper synthesis, the logic alphabet of this invention is a combined brain, better yet, a unified brain notation. That is to say, the logic alphabet simultaneously and structurally incorporates both sides of the algebraic and geometric extremes, both the letter and the shape aspects of letter-shapes, thereby making it possible to design a symmetry positional notation that is phonetic-iconic.
Unlike the PWR and Polish systems, the logic alphabet of this invention has iconicity. That is to say, the visual meaning assigned to each letter-shape is a matching image of its logical meaning. Furthermore, by carefully adapting iconicity to another feature called frame consistency, the logic alphabet in accordance with this invention has eusymmetry. That is to say, also unlike the PWR and Polish systems, and unlike the McCulloch system as heretofore practiced, each letter-shape in the logic alphabet has been carefully selected so that, as a matter of good symmetry, it uniformly participates in one, all-common set of geometric orientations, namely, the same system of flips and rotations.
Another important and unique characteristic of the logic alphabet of this invention is that each of the symbols has transformational facility. Logical operations are performed automatically because, unlike the currently used systems of PWR, Polish, and McCulloch, the physical operation of flipping and rotating acts upon the letter-shapes themselves. As a result, also unlike the PWR, Polish, and McCulloch systems, the transformational facility of this invention not only reduces significantly the need for abstract rules but also, when performing logical operations, greatly simplifies the nature of these rules.
In accordance with this invention and in the same act that establishes frame consistency, each letter-shape in the logic alphabet is assigned its logical meaning by carefully relating it to an all-common basic square, one that usually remains unwritten, but even more important, one that is always retained at the mental level. In contrast, the PWR and Polish systems employ no such frame of meaning, and the X-frame in the McCulloch system is not retained at the mental level but, instead, must be written each time a symbol is used. As described below, as other features will show, the unwritten, mental square constitutes a fundamental part of the logic alphabet of this invention.
In addition, to facilitate understanding of the deep commonalities between logic and mathematics, it is a feature of this invention to adapt the unwritten, mental square so that it contains the traditional order assigned to the x-y co-ordinates of analytic geometry. To accomplish this in the logic alphabet of this invention, the TT, TF, FT, and FF compartments of an ordinary Venn diagram are contracted to the smallest possible size, namely, set-regions reduced to the size of points, which thereby establishes the limit case of state space reduction. Next, treated as elements having point set size, the corners of the frame-consistent basic square are coded to represent the entries in the ordinary truth table for two sentences (TT, TF, FT, FF), that is, coded so that they are placed in Cartesian order. Thus, in the same act that establishes not only frame consistency but also iconicity and eusymmetry, the patterns of true and false at the four coded corners are arranged to match the patterns of plus and minus in the four quadrants of the x-y axes, as typically employed in analytic geometry.
The logic alphabet of this invention has a good thinkwrite ratio. This condition follows from the way in which the above described features are combined with the delicate balance contributed by the mental role and the unwritten aspect of the all-common basic square. On the one hand, unlike prior notations that expect the user to think too much, such as the PWR and Polish systems, an operator employing the logic alphabet can avoid many unnecessary, abstract mental manipulations and the memory work that goes with them. On the other hand, unlike prior notations that expect the user to write too much, such as the McCulloch system, the operator does not engage in unnecessary, repetitious work-writing of the all-common reference frame when employing the new symbols. Consequently, for each symbol of the logic alphabet of this invention, as it participates in the good think-write ratio, the basic square in thought on the one hand and the minimal letter-shape in writing on the other are evenly weighted and greatly reduced man-sign components of the total symbolic act-process.
A consequence of the several features of this invention is that the logic alphabet, unlike the PWR and McCulloch systems, has typographical potential. This statement is conservative: twelve of the sixteen lower-case letter-shapes are already included on the ordinary keyboard of a standard typewriter. Adding only four symbols to the keyboard makes the logic alphabet, like the Polish system, completely typographical. These symbols ( , , , ), old shapes in new positions, are obtained when the c-letter is flipped from left to right and when the h-letter is flipped both ways and rotated through a half-turn.
Another consequence of the several features of this invention relates to the use of parentheses. As a matter of prior practice, the PWR system is parenthesis-bound and the Polish system is parenthesis-free. In contrast, in keeping with the mental economy of the moment, the logic alphabet of this invention can be used with or without parentheses, even in a mixed way, if the operator so chooses.
Furthermore, a unique and important advantage of the logic alphabet as a notational system, in accordance with this invention, is that it facilitates the use of a large family of physical embodiments or models that can be employed in computing, teaching, and demonstrating standard logical operations. Finally, exceptional far beyond the prior notational systems discussed above, the physical embodiments of this invention can be displayed with great clarity, both visually and tactually, thereby not only fostering learning at the sensorimotor level but also making explicit in an elegant and aesthetic manner the underlying structures that inhabit elementary symbolic logic.
In summary, in reference to the sixteen binary connectives, the logic alphabet of this invention is a unique notation that has a systematically pursued and a carefully combined set of special properties. It is complete, geometric, and phonetic; it has iconicity, frame consistency, and eusymmetry; its unwritten, mental basic square is placed in Cartesian orientation; it is based on lower-case letter-shapes that are symmetry positional. It has transformational facility, typographical potential, and a good think-write ratio; if preferred, it is parenthesis-free. It consists of a society of symbols for which manipulatory structure has been designed to reflect logical structure. Especially, as a primary consequence of the foregoing, it easily lends itself to the construction of a large family of physical models which in turn reflect both the matching manipulatory structure and the underlying logical structure.
The following TABLE I is a comparison of the major ways of expressing the sixteen binary connectives, including the three above-noted prior systems and the Logic Alphabet of this invention.
TABLE I __________________________________________________________________________ Truth Logic No. Table In Words PWR Polish McCulloch Alphabet __________________________________________________________________________ 1 FFFF Contradiction Contradiction O o 2 FFFT Not-A and Not-B .about. A .multidot. .about. B X P 3 FFTF Not-A and B .about. A .multidot. B M b 4 FTFF A and Not-B A .multidot. .about. B L q 5 TFFF A and B A .multidot. B K d 6 TTFF A A I c 7 TFTF B B H u 8 TFFT A equivalent B A .ident. B E s 9 FTTF A or else B A B J z 10 FTFT Not-B .about. B G n 11 FFTT Not-A .about. A F 12 FTTT Not-A or Not-B A/B D h 13 TFTT if A, then B A B C 14 TTFT if B, then A B A B 15 TTTF A or B A B A 16 TTTT Tautology Tautology V x __________________________________________________________________________